2024/01/14 by Alaa Abouhalaka, Abouhalaka, Alaa
Mathematics · #16D25 #16D99 #16N60 #16W99 #Algebraic structures and combinatorial models #Commutative Algebra and Its Applications #FOS: Mathematics #Rings and Algebras (math.RA) #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2401.07270
openalex publication_date 2024/01/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let S be an m-system of a ring R, and P a submodule of a right R-module M. This paper, presents the notion of S-prime submodule and provides some properties and equivalent definitions. We define S-multiplication right module, and prove that in multiplication (S-multiplication) right R-module M, the ideal (P :R M) is a right S-prime ideal of R if and only if P is an S-prime submodule of M. Moreover, we give an S-version of prime avoidance lemma. Furthermore, we define S-finite and S-Noetherian right modules following the definitions in [1]. We prove that a multiplication finitely generated right R-module M is S-Noetherian if (N :R M) is an S-prime ideal of R, for all submodules N of M. In addition, we give some examples of right S-Noetherian rings.